11 problems

Let $\Omega\subset\mathbb{R}^N$ be the domain and let $u\in W_0^{1,2}(\Omega)$ be a stable weak solution of the elliptic problem considered in the paper. Assume that $f$ is positiv…

de Figueiredo–Lions–Nussbaum conjecture. This condition is not necessary for the uniform a priori bounds established for positive solutions, although it is essential in their proof…

Let $\Omega$ be the domain and consider the coupled elliptic system with positive exponents $p_1,p_2$, where the critical hyperbola condition is … A positive weak solution is a wea…

Uniform a priori estimate conjecture. There exists $C(M,g,\delta)>0$ such that

Donald's conjecture. For any $k\geqslant 0$, one can bound $\|\tilde{\omega}\|_{C^k(M,g)}$ by a constant depending only on $k$ and on bounds for $\sigma$, $\omega$, and $(M,J)$, wh…

A backward stochastic differential equation (BSDE) with terminal time $\tau$, terminal condition $be$, and generator $f$ is written as $\operatorname{BSDE}^{\tau}(\xi,f)$; its data…

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain, and let $(u,v)$ be a classical solution of the Lane–Emden system with exponents $p,q\geq 1$ satisfying … for some…

Optimal exponent conjecture. The best exponent $a$ is equal to $1$, and

Let $(u,p,u_0,0,T,L)$ be a smooth periodic, homogeneous normalised-pressure solution, with $0<T<T_0<\infty$ and periodic data norm ${\mathcal H}^1(u_0,0,T,L)\leq A<\infty$. A prior…

Let $(u,p,u_0,0,T)$ be a smooth homogeneous $H^1$ solution, with $\|u_0\|_{H^1_x(\mathbf{R}^3)}\leq A<\infty$. A priori global homogeneous $H^1$ bound. There exists a function…

Let $(u,p,u_0,f,T,L)$ be a smooth periodic, normalised-pressure solution, with periodic $H^1$ data norm ${\mathcal H}^1(u_0,f,T,L)$ finite. A priori periodic $H^1$ bound. There exi…